Spectrum of (, q)-fuzzy Prime h-ideals of a Hemiring

Transcription

1 World Appled Scences Journal 17 (12): , 2012 ISSN IDOSI Publcatons, 2012 Spectrum of (, q)-fuzzy Prme h-deals of a Hemrng 1 M. Shabr and 2 T. Mahmood 1 Department of Mathematcs, Quad--Azam Unversty, Islamabad, Pakstan 2 Department of Mathematcs and Statstcs, Internatonal Islamc Unversty, Islamabad, Pakstan Abstract: In ths paper we defne (, q)-fuzzy prme h-deal, (, q)-fuzzy semprme h deal. We also construct spectrum of (, q)-fuzzy prme h-deals n two dfferent ways. In the last secton we also dscuss mlpcaton-based (, q)-fuzzy prme h-deals mathematcs subject classfcaton: 16Y60 08A72 03G25 03E72 Key words: (, q)-fuzzy prme h-deal (, q)-fuzzy semprme h-deal spectrum of (, q)- fuzzy prme h-deal INTRODUCTION Semrngs, ntroduced by Vandver [1] n 1934., appears n natural way for studyng optmzaton theory, graph theory, theory of dscrete event dynamcal systems, matrces, determnants, generalzed fuzzy computaton, theory of automata, formal languages theory, codng theory, analyss of computer programmes. Hemrngs, whch are semrngs wth commutatve addton and absorbng addtve dentty, are used n some applcatons to the theory of automata, the theory of formal languages and n computer scences [2-5]. Ideals of semrngs play an mportant role n the structure theory of semrngs. However, n general, they do not concde wth the usual rng deals. So many results n rng theory have no analogues n semrngs usng only deals. In order to overcome ths dffculty n [6] Henrksen defned a more restrcted class of deals n semrngs, called k-deals, wth the property that f a semrng R s a rng, then a comple n R s a k-deal f and only f t s a rng deal. Another more restrcted class of deals n hemrngs, called now h-deals, has been gven and nvestgated by Izuka [7]. In 1965, Zadeh [8] ntroduced the concept of fuzzy set. Snce then many efforts have been made to use ths concept n algebrac structures. In ths connecton, Rosenfeld [9] ntroduced the noton of fuzzy subgroups. In [10] J. Ahsan et. al ntated the study of fuzzy semrngs. The notons of "belongngness ( )" and "quasconcdence (q)" of fuzzy ponts and fuzzy sets proposed and dscussed n [11, 12]. In [13], Zhan and Dudek ntroduced the concept of prme fuzzy h-deals of a hemrng and establshed some nterestng results. In [14], Kumbhojkar addressed the questons rased by Zhan and Dudek, redefned prme fuzzy h-deals, defned semprme fuzzy h-deals of a hemrng and constructed spectrum of prme fuzzy h-deals. Snce prme spectrum on the set of prme deals of a commutatve hemrng wth unty plays an mportant role n commutatve algebra, algebrac geometry, lattce theory and semprme deals of hemrngs determne the topology. So keepng n vew all these facts n ths paper we etend the deas of Kumbhojkar and defne (, q)-fuzzy prme h-deal, (, q)-fuzzy semprme h-deal. We also construct spectrum of (, q)-fuzzy prme h-deals n two dfferent ways. In the last secton we also dscuss mplcaton-based (, q)-fuzzy prme h-deals. PRELIMINARIES For basc defntons we refer to [4, 14]. A fuzzy subset ƒ of a unverse X s a functon ƒ: X [0,1]. A fuzzy subset ƒ of X of the form t (0,1] f y= f ( y) = 0 f y s called the fuzzy pont wth support and value t and s denoted by t. Furthermore for t (0,1], level subset of ƒ s denoted and defned by U(f,t) = { X:f() t}. Correspondng Author: T. Mahmood, Department of Mathematcs and Statstcs, Internatonal Islamc Unversty, Islamabad, Pakstan 1815

2 World Appl. Sc. J., 17 (12): , 2012 In [12] Pu and Lu defned and dscussed t αƒ, α,q, q, q. A fuzzy pont t s sad where { } to belong to (resp. quas-concdent wth) ƒ, wrtten t resp. f + t > 1. as t ƒ ( resp. qf t ), f f qf ( resp. qf) t ( resp. f and qf ) t means that t ƒ or t qƒ t t. To say that t α f means that t αƒ does not hold. For any two fuzzy subsets ƒ and g of X, ƒ g means that, for all X, ƒ() g(). For two fuzzy subsets ƒ and g of X, ƒ g and ƒ g wll mean the followng fuzzy subsets of X 2 (1 a) t f t qf Theorem: For a fuzzy subset ƒ of R the followng are equvallent: 2 (1 a) t f t qf (1 b) 2 f() f 0.5 for all R and t (0,1]. Corollary: A fuzzy h-deal ƒ of R s an (, q)-fuzzy semprme h-deal of R f and only f t satsfes (1 b). for all X. (f g)() = f() g() (f g)() = f() g() Lemma: A fuzzy h-deal ƒ of R s an (, q)-fuzzy semprme h-deal of R f and only f for all t (0,0.5], the non-empty level subset U(ƒ,t) s ether semprme h- deal of R or s R tself. (, q)-fuzzy PRIME AND (, q)-fuzzy SEMIPRIME H-IDEALS From now to onward throughout R wll denote a commutatve hemrng wth 1. Defnton: A fuzzy h-deal ƒ of R s sad to be an (, q)-fuzzy prme h-deal of R f t s non constant and for all,y R and t (0,1] (1a) (y) t f t qf or yt qf Theorem: For a fuzzy subset ƒ of R the followng are equvallent: (1a) (y) t f t qf or yt qf (1b) f() f(y) f(y) 0.5 for all,y R and t (0,1]. Corollary: A fuzzy h-deal ƒ of R s an (, q)-fuzzy prme h-deal of R f and only f t satsfes (1b). Lemma: A non-empty subset P of R s semprme h- deal of R f and only f C P s an (, q)-fuzzy semprme h-deal of R. Remark: Every (, q)-fuzzy prme h-deal of R s (, q)-fuzzy semprme h-deal of R, however converse s not true n general. Eample: Let N 0 = {0} N and p 1, p 2, p 3, be the dstnct prme numbers n N 0. If J 0 = N 0 and J l = p 1 p 2 p 3 p l N 0, l = 1,2,3,, then J 0 J 1 J 2 J n J n+1 As every non-zero element of N 0 has unque prme factorzaton, J l s a semprme h-deal for l = 2,3, but not a prme h-deal. Then for such values of l, by Lemma 3.10, C s an (, q)-fuzzy semprme h- l J deal of R, but by Lemma 3.5, C l s not an (, q)- J fuzzy prme h-deal of R. In our net dscusson Ω denotes the set of all (, q)-fuzzy prme h-deals of R, also we assume for every f Ω, ƒ(0) = 1. Lemma: A fuzzy h-deal ƒ of R s an (, q)-fuzzy prme h-deal of R f and only f for all t (0,0.5], the non-empty level subset U(f,t) = {:f() t} s ether prme h-deal of R or s R tself. Lemma: A non-empty subset P of R s prme h-deal of R f and only f C P s an (, q)-fuzzy prme h-deal of R. Defnton: A fuzzy h-deal ƒ of R s sad to be an (, q)-fuzzy semprme h-deal of R f t s non constant and for all R and t (0,1]. SPECTRUM OF (, q)-fuzzy PRIME H-IDEALS We denote and defne Ω = {ƒ:ƒ s (, q)- fuzzy prme h-deal of R}. If λ Ω, then V( λ ) = {f Ω: λ f}. If λ= C {a}, where a R, then V( λ ) = V(a) = {f Ω :C{a} f} = {f Ω :f(a) = 1} Further r() h λ = {f Ω: λ f} s called (, q)- fuzzy h-prme radcal of λ. Further 1816

3 World Appl. Sc. J., 17 (12): , 2012 V( λ ) =Ω V( λ ) = {f Ω : λ> f} Lemma: r h (λ) s (, q)-fuzzy semprme h-deal contanng λ. Proof: Proof s straghtforward, hence omtted. Lemma: If λ,µ are fuzzy subsets of R. Then Proof () As V(o) =φ and V( R ) =Ω, where R Ω, defned by R() = 1, for all R. So φω IΩ,. () Let {V( λ): I} be an arbtrary collecton of elements of I(Ω). To prove IV( λ ) s n I(Ω). () λ µ mples V( µ ) V( λ ) () V( λ) V( µ ) V( λ µ ) () If A and B are h-deals of R, then V(C A) V(C B) = V(C A B) (v) If η s (, q)-fuzzy h-deal of R generated by λ and r h (η) s (, q)-fuzzy h-prme radcal of η. Then V( λ ) = V( η ) = V(r h ( η )). (v) If { λ : Λ } s a famly of fuzzy subsets of R, then V( λ { : Λ }) = λ { : Λ } (v) If A R, then V(C A) = {V(a): a A} (v) V(a) V(b) = V(ab), for all a,b R. Proof () Let f V( µ ). Then µ f λ µ f f V( λ ). () As λ µ λ and λ µ µ, by () V( λ) V( λ µ ) and V( µ ) V( λ µ ) V( λ) V( µ ) V( λ µ ). () By usng Lemma 2.5 and () V(C A) V(C B) V(C A B). Now let f V(C A B). Whch mples CA B f f() = 1 for all A B. Then ƒ() = 1 for all A or ƒ() = 1 for all B CA f or CB f f V(C A) or f V(C B) f V(C A) V(C B). Hence V(C A) V(C B) = V(C A B). (v) Frst we prove V( λ ) = V( η ). Let Γ be the class of all (, q)-fuzzy h-deals of R contanng λ. Then η= Γ. Then by the fact λ f η f, result follows mmedately. Now we prove V( η= ) V(r ( η )). Let Γ be the class of all (, q)- h fuzzy prme h-deals of R contanng η. Then r() η = Γ. Then by the fact r() η f η f, h result follows mmedately. (v) Let f λ { : Λ }. Then f λ for all Λ λ f for all Λ λ { : Λ} f f V( λ { : Λ }) λ { : Λ} V( λ { : Λ }). Workng backward we get the desred result. (v) Follows drectly from (v). (v) Straghtforward. Theorem: The set I( Ω ) = {V( λ): λ Ω } forms a topology on the set Ω. h Note that equal to ƒ(), then 1817 V( ) { f : f} { f : k I such that f} λ = Ω λ > = Ω λ > λ1( a1) λ2( a 2)... λ = λ1( b1) λ2( b 2)... I + a1+ a = b1 + b where a,b R for all and also only a fnte number of the as and bs are not zero. Snce λ (0) = 1, therefore we are consderng the nfmum of a fnte number of terms because 1 s are effectvely not beng consdered. Now, f for some k I, λ k >ƒ, then there ests R such that λ k ()>ƒ(). Consder the partcular epresson for whch a k =, b k = 0 and a = b = 0 for all k. We see that λ k () s an element of the set whose supremum s defned to be ( λ ). k. Ths mples I Thus ( λ ) λ > f ( λ ) > f that s λ > f. Hence λ k >ƒ for some k I mples λ > f. Conversely, suppose that ests R such that ( λ ) > k then there λ > f f. λ1 a1 λ2 a 2... > f λ a a... z b b... z 1 b1 λ2 b = λ1 a1 λ2 a 2... λ1 b1 λ2 b 2... > f ( ) ( a1 a 2... b1 b = + + beng the correspondng breakup of, where only a fnte number of as and bs are not zero.) Now on the contrary f we suppose that (*) does not hold, that s f all the elements of the set (whose supremum we are takng) are ndvdually less than are

4 World Appl. Sc. J., 17 (12): , 2012 λ1( a1) λ2( a 2)... λ = λ a 1( b1) λ2( b 2)... + I 1+ a z= b1+ b z f whch s a contradcton. Thus (*) holds. Thus, Let and λ1( 1) λ2( 2) λ1( 1) λ2( 2) ( 1) ( 2) ( 1) ( 2) a a... b b... > f f a f a... f b f b... ( a ) ( a )... ( b ) ( b )... ( ) λ1 1 λ2 2 λ1 1 λ2 2 =λ l l where 1 l ( 1) ( 2) ( 1) ( 2) = ( l) f a f a...f b f b... f So, l( l) f( l) λ > t follows that λ l >ƒ for some l I. Hence λ > f mples that λ l >ƒ for some l I. Hence the two statements some l I are equvalent. Hence As { } V( λ ) = f Ω : λ > f λ Ω and λ l >ƒ for λ > f = f Ω : λ > f = V( λ) I I thus, IV( ) () Let V( ),V( ) λ I Ω. λ1 λ2 IΩ. Proof: As for a R,V(a) = {f Ω :f(a) = 1}, so V(a) = {f Ω :f(a) 1}. Now as V (1) = Ω, so {V(a):a R} =Ω. Also by usng (v) of Lemma 4.2, V(ab) = V(a) V(b) for all a,b R Thus B s base for some topology T on Ω. Now let U T,-then for some A R, U = {V(a):a A} = Ω { V(a): a A} =Ω {V(a): a A} =Ω V(C ) (by (v) of Lemma 4.2) = V(C ) A Further as V(C A) = V(C A ) = V(r h(c A )). Hence T s completely determned by (, q)-fuzzy semprme h-deals of R. IMPLICATION-BASED (, PRIME H-IDEALS A q)-fuzzy Fuzzy propostonal calculus s an etenson of the Arstotelan propostonal calculus. In fuzzy propostonal calculus the truth set s taken [0, 1] nstead of {0, 1}, whch s the truth set n Arstotelan propostonal calculus. In fuzzy logc some of the operators lke,,, can be defned by usng truth tables. One can also use the etenson prncple to obtan the defntons of these operators. In fuzzy logc the truth value of a fuzzy proposton λ s denoted by [λ]. In the followng we gve fuzzy logc and ts correspondng set theoretcal notatons, whch we wll use n the paper hereafter. [ λ=λ ] (), [ λ ] = 1 λ () [P Q] = mn{[p],[q]}, [P Q] = ma{[p],[q]} [P Q] = mn{1,1 [P] + [Q]} [ P()] = nf[p()] P f and only f [P] = 1. Then { } V( λ1) V( λ 2 ) = f Ω : λ 1 > f and λ 2 > f = V( λ1 λ2) Hence t follows that I(Ω) forms a topology on the set Ω. Operator name Early Zadeh Lukasewcz Standard star (Godel) Contraposton of Godel Theorem: If a R and V(a) =Ω V(a). Then Gans-Rescher B = {V(a):a R} s a base for a topology T on Ω. Then Kleen-Denes open sets of ths topology are V(C A ),A R. Further Goguen ths topology s completely determned by (, q)- fuzzy semprme h-deals of R Defnton of the operator I m (,y) = ma{1,mn{,y}} I a (,y) = mn{1,1 + y} 1 f y I g(,y) = y f y< 1 f y I cg(,y) = 1 f y < 1 f y I gr(,y) = 0 f y< I b(,y) = ma{1,y} 1 f y I gg(,y) = y f y<

5 World Appl. Sc. J., 17 (12): , 2012 A functon I:[0,1] [0,1] [0,1] s called fuzzy mplcaton f ts monotonc wth respect to both varables separately and fulflls the bnary mplcaton truth table By monotoncty of I I(1,0) = 0,I(0,0) = I(0,1) = I(1,1) = 1 I(0,) = I(,1) = 1 for all [0,1]. There have been defned many mplcaton operators n [15]. We consder n the followng some mportant Implcaton Operators: In the followng defnton, we consder the mplcaton operators n Lukasewcz system of contnuous-valued logc. Defnton: A fuzzy set λ of R s called a fuzzfed left (rght) prme h-deal f t satsfes: () For any,y R, [ λ] [y λ] [+ y λ ], () For any,y R, [y λ] [y λ]( resp. [ λ] [y λ ]), () For any,y R, [y λ] [ λ] [y λ ], (v) For any a,b,,z R wth + a+ z = b + z, [a λ ] [b λ ] [ λ ]. A fuzzy set λ of a hemrng R s called a fuzzfed prme h- deal f and only f t s both fuzzfed left and rght prme h-deal of R. Corollary: Let I be an mplcaton operator, t (0,1] s a fed number and λ be a fuzzy set of R. Then λ s a t-mplcaton based fuzzy prme h-deal of R f and only f the followng condtons hold: () For any,y R,I( λ() λ(y), λ (+ y)) t, () For any,y R,I( λ(), λ(y)) t( resp. I( λ(y), λ(y)) t), () For any,y R,I( λ(y), λ() λ(y)) t, (v) For any a,b,,z R wth + a + z = b+ z,i( λ(a) λ(b), λ()) t. Theorem (1) Let I = I gr. Then λ s a 0.5-mplcaton-based fuzzy prme h-deal of R f and only f λ s a fuzzy prme h-deal of R. (2) Let I = I g. Then λ s a 0.5-mplcaton-based fuzzy prme h-deal of R f and only f λ s an (, q)- fuzzy prme h-deal of R. Proof: (1) Let us assume λ be a 0.5-mplcaton-based fuzzy prme h-deal of R. Then () For any,y R,I gr ( λ() λ(y), λ (+ y)) 0.5, () For any,y R,I gr( λ(), λ(y)) 0.5( resp. I( λ(y), λ(y)) 0.5), () For any,y R,I gr( λ(y), λ() λ(y)) 0.5, (v) For any a,b,,z R wth + a + z = b+ z,i ( λ(a) λ(b), λ()) 0.5. gr In [23] the concept of t-tautology s gven, that s tp f and only f [P] t. Now we defne the mplcaton based fuzzy h-deal: Defnton: A fuzzy set λ of R s called a t-mplcatonbased, t (0,1] s a fed number, left (rght) prme h- deal f t satsfes: () For any,y R, t[ λ] [y λ] [+ y λ ], () For any,y R, t[y λ] [y λ ]( resp. [ λ] [y λ ]), () For any,y R, t[y λ] [ λ] [y λ ], (v) For any a,b,,z R wth + a + z = b + z, [a λ] [b λ ] [ λ ]. A fuzzy t set λ of a hemrng R s called a t-mplcatonbased, t (0,1] s a fed number, prme h-deal f and only f t s both t-mplcaton-based left and t-mplcaton-based rght prme h-deal of R Now from () the only case whch s possble s that λ ( + y) λ() λ (y). Smlarly other condtons can be proved for λ to be prme fuzzy h-deal of R. Converse of the Theorem s straghtforward. (2) Let us assume λ be a 0.5-mplcaton-based fuzzy prme h-deal of R. Then () For any,y R,I g( λ() λ(y), λ ( + y)) 0.5, () For any,y R,I g( λ(), λ(y)) 0.5( resp. I( λ(y), λ(y)) 0.5), () For any,y R,I g( λ(y), λ() λ(y)) 0.5, (v) For any a,b,,z R wth + a+ z= b+ z,i ( λ(a) λ(b), λ()) 0.5. g Now () mples λ ( + y) λ() λ (y) or λ() λ (y) >λ ( + y) 0.5. Then ( y) mn{ () (y) 0.5}. λ + λ λ () and (v) can be proved smlarly.

6 World Appl. Sc. J., 17 (12): , 2012 Net () mples λ() λ(y) λ (y) or λ (y) >λ() λ(y) 0.5. Then λ() λ(y) λ(y) 0.5. Hence λ s an (, q)-fuzzy prme h-deal of R. Converse s straghtforward. REFERENCES 1. Vandver, H.S., Note on a smple type of algebra n whch cancellaton law of addton does not hold. Bull. Amer. Math. Soc., 40: Glazek, K., A gude to lterature on semrngs and ther applcatons n mathematcs and nformaton scences: Wth complete bblography, Kluwer Acad. Publ. Nederland. 3. Wechler, W., The concept of fuzzness n automata and language theory. Academc verlog, Berln. 4. Golan, J.S., Semrngs and ther applcatons, Kluwer Acad. Publ. 5. Mordeson, J.N. and D.S. Malk, Fuzzy Automata and Languages. Theory and Applcatons, Computatonal Mathematcs Seres, Chapman and Hall/CRC, Boca Raton. 6. Henrksen, M., Ideals n semrngs wth commutatve addton. Amer. Math. Soc. Notces, Izuka, K., On Jacobson radcal of a semrng. Tohoku Math. J., 11: Zadeh, L.A., Fuzzy Sets. Informaton and Control, 8: Rosenfeld, A., Fuzzy groups. J. Math. Anal. Appl., 35: Ahsan, J., K. Safullah and M.F. Khan, Fuzzy Semrngs. Fuzzy Sets Syst., 60: Mural, V., Fuzzy ponts of equvalent fuzzy subsets. Inform. Sc., 158: Pu, P.M. and Y.M. Lu, Fuzzy topology I, neghborhood structure of a fuzzy pont and Moore-Smth convergence. J. Math. Anal. Appl., 76: Zhan, J. and W.A. Dudek, Fuzzy h-deals of hemrngs. Inform. Sc., 177: Kumbhojkar, H.V., Spectrum of prme L- fuzzy h-deals of a hemrng, Fuzzy Sets and Systems. do: /j.fss Nguyen, H.T. and E.A. Walker, A frst course n fuzzy logc. Chapman and Hall/CRC, Boca Raton. 1820